9th February 2021

Poisson Log-Normal Distributed Random Numbers

Task at hand: Generate random numbers which follow a lognormal distribution, but this drawing is governed by a Poisson distribution. I.e., the Poisson distribution governs how many lognormal random values are drawn. Input to the program are \(\lambda\) of the Poisson distribution, modal value and either 95% or 99% percentile of the lognormal distribution.

From Wikipedia's entry on Log-normal distribution we find the formula for the quantile \(q\) for the \(p\)-percentage of the percentile \((0<p<1)\), given mean \(\mu\) and standard deviation \(\sigma\):

$$ q = \exp\left( \mu + \sqrt{2}\,\sigma\, \hbox{erf}^{-1}(2p-1)\right) $$

and the modal value \(m\) as

$$ m = \exp\left( \mu - \sigma^2 \right). $$

So if \(q\) and \(m\) are given, we can compute \(\mu\) and \(\sigma\):

$$ \mu = \log m + \sigma^2, $$

and \(\sigma\) is the solution of the quadratic equation:

$$ \log q = \log m + \sigma^2 + \sqrt{2}\,\sigma\, \hbox{erf}^{-1}(2p-1), $$


$$ \sigma_{1/2} = -{\sqrt{2}\over2}\, \hbox{erf}^{-1}(2p-1) \pm\sqrt{ {1\over2}\left(\hbox{erf}^{-1}(2p-1)\right)^2 - \log(m/q) }, $$

or more simple

$$ \sigma_{1/2} = -R/2 \pm \sqrt{R^2/4 - \log(m/q) }, $$


$$ R = \sqrt{2}\,\hbox{erf}^{-1}(2p-1). $$

For quantiles 95% and 99% one gets \(R\) as 1.64485362695147 and 2.32634787404084 respectively. For computing the inverse error function I used erfinv.c from lakshayg.

Actual generation of random numbers according Poisson- and lognormal-distribution is done using GSL. My program is here: gslSoris.c.

Poisson distribution looks like this (from GSL documentation): Poisson distribution

Lognormal distribution looks like this (from GSL): Lognormal distribution